A sequence $\{f_{n}\}_{n\in I}$ is a frame for a separable Hilbert space $H$ if there exists $0<A\leq B<\infty$ such that $$ A\|f\|^{2} \leq \sum_{n\in I}|\langle f,f_{n}\rangle|^{2}\leq B\|f\|^{2}$$ for all $f\in H$. If the sequence only satisfying the upper bound it is called a Bessel sequence.
Now my questin is: if a given sequence is Bessel sequence but not a frame, what does this mean?
My guess is that: there exists (a non-zero) $f\in H$ such that $$ A\|f\|^{2} > \sum_{n\in I}|\langle f,f_{n}\rangle|^{2} $$ for all $A$.
But I'm not sure if this is correct! Any help is appreciated!